Conjugacy Classes
By Jordan normal form, matrices are classified up to conjugacy (in GL(n,C)) by eigenvalues and nilpotence (concretely, nilpotence means where 1s occur in the Jordan blocks). Thus elements of SL(2) are classified up to conjugacy in GL(2) (or indeed SL±(2)) by trace (since determinant is fixed, and trace and determinant determine eigenvalues), except if the eigenvalues are equal, so ±I and the parabolic elements of trace +2 and trace 2 are not conjugate (the former have no off-diagonal entries in Jordan form, while the latter do).
Up to conjugacy in SL(2) (instead of GL(2)), there is an additional datum, corresponding to orientation: a clockwise and counterclockwise (elliptical) rotation are not conjugate, nor are a positive and negative shear, as detailed above; thus for absolute value of trace less than 2, there are two conjugacy classes for each trace (clockwise and counterclockwise rotations), for absolute value of the trace equal to 2 there are three conjugacy classes for each trace (positive shear, identity, negative shear; and the negatives of these), and for absolute value of the trace greater than 2 there is one conjugacy class for a given trace.
Read more about this topic: SL2(R), Classification of Elements
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